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M iXX Cg 0XX I n 7k 7c 71XX AckXXFkXX/Mi,kXXA(XXXTXMkXXN100XX))<4 <T(y=0)=T_jHORZ25,HORZ35({dT}over{dy})_{y=L}=HORZ150 XXTXXv(XXyXXXX0XX)XX.XXTjXXq,XX(&X7XXMfdTXXYYdyXX7)yyLXXMXX0 TQ_j=(T_infinityT_j)sqrt{{k_{tc}A_{tc}}overR}tanhLover{sqrt{k_{tc}A_{tc}R}} XXFQjXXFXX7F(XXyFTXX9FXXFT&jXXJF)X>+XX+TXXSXXSXXnSXX/SXXrR XJpXX`ktcXXAtcXXRXXFtanh XhpXXLX~PXXkE5tcXXA 5tcXX{ R R=1over{h2pir}  *#XXXX  ~XX{~~ + Figure  1   ݀Measurementlocation#~{~[##XX~<# XXRXX) XXXf1XXYh2XXY!XX^Yr  *#}XXXX  }~X}X + Figure  2   #}XX}~:#Ԁ~X9ZXX}X~:-X~X9ZCrosssectiononehalfof315kVAexplosionproof  transformer:1enclosure,2core,3lowvoltagecoils(LV1,LV2),4barrier,5highvoltagecoils(HV1,HV2).#}~-~:##}XX}~#Ԁ(  Z 6Times New Roman Regular XXTXX(XXyXX'XX0XX)XXEXXT2jXXV,XX(&XXXVdTXXGdyXX)yRLXXXXd0(  Z 6Times New Roman Regular) `CG Times p *#XXXX  ~XX + Figure  3   #XX~:#Ԁ& &XX~&& Nodalnetworkforonehalfofthetransformer.Nodes:1,15coresurface,PhaseB:2,3,4LV1coil,  5,6,7LV2coil,8barrier,9,10,11HV1coil,12,13,14HV2coil,PhaseC:nodes1528,29enclosedair,30,31innersurfaceoftheenclosure,32,33outersurfaceoftheenclosure,34surroundingair.#& &~#(  Z 6Times New Roman RegularT@TTTTT333333?TQ?TT@T?T̬_@T$~?GT(\?Tffffff)@T|\@T@T?T`@TA`"?GT(\?T333333*@Tףp= \@T@T@T a@THP?GT(\?T/@T(\"^@T @T@T`@T_vO?GT333333?T333333/@T ףp=]@T"@T?T`@TH}?GT333333?T+@TQ\@T&@T?T_@TS!uq?GT(\?T333333*@T(\¥\@T(@T@T^@Tݓ?GT(\?T&@T{G[@T,@T@Tfffff^@T_)Ǻ?GTQ?T%@TGzd[@TABLE AT /Q_j=rk_w(T_jT_w){Q}^*+hpir^2(T_infinityT_w) XXxQ7jXXxXXHxrk7wXXKx(XXxT7jXX&xXXxT7wXXkx)XXxQ=XXxXXxhXXix!XXxrS2XXx(XXxTG7XXxXX xT 7wXX x)(  Z 6Times New Roman Regular 5@!TRW6&A43'6&Transparency A43'TXXXX  v   XXB*c*k* 5@TRW6&A4W6&A46&Transparency A46&Transparency A4T      TRW6&A4W6&A4 6&A46&Transparency A4TAnalysisofsurfacetemperaturemeasurementerrorsincomplex v apparatabyapplicationofnetworkanalogy#XX  #   & &XXDavorZvizdic(Zagreb),IvanGalaso(Zagreb)#XX&& Ӯ# $    & &XX0  Thispaperdescribesthemethodologyforestimationofsurfacetemperaturemeasurementerrorsin  \ Мcomplexapparatawhere & &&& simultaneousheattransferbyconduction,convectionandradiationtakesplace.  2 #& && & #Thecomplexthermalsystemismodeledbyapplicationofnetworkanalogywithsurfacemeasurement ~ errorsmodeledwithemphasisonthermocoupleinducedlocalheatsinkandconvectivestraylooses.Thescopeofthemethodispresentedbycalculationofsurfacemeasurementerrorsoninnersurfacesofexplosionprooftransformer.  # # 1ӜӀ̜0  & &&&   0  DerArtikelbeschreibtdieMethodologiederBestimmungdesFhlersbeiderOberflchen A  temperaturmessunginkomplexenApparatenwoeingleichzeitigerWrmebergangdurchWrmeleitung,KonvektionundStrahlungstattfindet.DaskomplexethermischeSystemwirdmittelsderNetzanalogiemodeliert.DieFhlerdiewegen#& && & ` # & &&& ԀderrtlichenWrmesenkeundkonvektiverStreungverlustedurchdie M EinfhrungdesThermoelementesentstehen,werdenbercksichtigt.DieMethodeistamBerechnungbeispieldermessungfhleranderinnerenOberflcheeinesexplosionsicherenTransformatorsdargestellt.#& && & #ԛo # #  Q&&  L1  Introduction #& &Q # Q XX&& L  Thepowerapparataandcomponentslikepowermotors,transformers,switchinggear,airconditioning & units,refrigeratingunitsetc.arebecomingmorecompactandyetmorepowerfulandcomplex.Thisresultsineverincreasingcomponentdensityandincreasingrateofheatdissipationoftenleadingtoininsufficientcooling.Sincetemperatureriseinfluencecomponentlifetime,measurementofsurfacetemperaturesandconsequentlyestimationandanalysisofsurfacetemperaturemeasurementerrorsisanecessarystepinexperimentalanddesignstages.Surfacemountedthermocouplesareusedasarulesinceelectricalconsiderationsorotherreasonsdonotpermitwallpenetration.Butsuchapparataaretocomplextobeeconomicallymodeledwithsingletransientcontinuummodel(FEM,FDM,CVM,etc.)andaretoocompacttobedissectedintoclearlydefined,"blackbox"subcomponents,towhichlargescalesystemanalyzingtechniquescouldbeapplied.Insuchcases,especiallywhensimultaneousheattransferbyconduction,convectionandradiationtakesplace,concentratedparametermodelingmethodcanprovideforthesolutionwiththeabilitytodirectlyutilizeexistingempiricalheattransferandclosedformsurfaceerrorestimationknowledge.W  #& &XX& #Q&& L2  Methodologyandgoverningequations#& &Q#  (!% XX&& L  Inthefirststeptheheattransferincomplexapparatusismodeledbymeansofnetworkanalogywith m)"& certainnodesstrategicallypositionedatthesurfaceswheretemperaturemeasurementerrorsaretobemeasuredandestimated.Propertiesofthenonlinearthermalsystemareconcentratedatnindividualnodalpointswithnthnodehavingsurroundingorreferencetemperature.Foreachoftheremainingn1nodesinsteadystatecondition,thesumofexchangedheatmustequalgenerated(sourceorsink)energyinunittime:jӜ<2K;7z|J%  p@E/ %a/ <=<2K;7z|J  p@E/ =/ <==2K;7z|J  p@E/" ]fB/"  ##   0W*. =WhereTi,Tkaretemperaturesoftheithandkthnodes,Ri,k=Rk,i=R(Ti,Tj)isheattransfer v resistancebetweenithandkthnode,Pi(Ti)ispowersource/sinkintensityatithnode.Heat b transferresistancesbetweentwonodesareingeneraldefinedas:>>Ӝ>2 !J:6z|'U% 0@E h%a >H>2 !J:6z|'U 0@E "hB ">HH2 !J:6z|'U 0@E `0pQ ` ##H̜ ##WherePi,k=Pk,iisheatexchangedbetweenrespectivenodesinunittime.Conductionheattransfer  t resistancebetweennodesiandjequals:2VFBz (7% 0@Xdddddddd@E8ddcp$ag ##    ##nwhere eisequivalentthickness,eisequivalentcoefficientofthermalconductivityandAeis r  equivalentarea,allofwhichcanbecalculatedaccordingtobasic9999[1,2]::::equivalentingrules.Convectionheattransferresistancebetweennodesiandkcanbecalculatedas:@2ln>\LHzx=7% p@Xdddddddd@Emdda6gq ##    ##@@o2 >\LHzx=7% p@Xdddddddd@oߜ@wherehisheattransfercoefficientusuallycalculatedfromempiricalformulas[1,??2]forvariouslyshapedbodiesandfluidflowconditions,orcanresultfromnestednumericalcalculationencompassingrelevantsmallerareaoftheapparatus.Radiationheattransferresistancebetweennodesiandkaredefinedas: XXXX2?@d[KGzx=% 0@Xdddddddd@E)8ddp$ag ##     ##LwherePRisheatexchangedbyradiationbetweennodesiandkinunittime. 2! LNodesexchangingheatbyradiationcanfrequentlybeapproximatedbyasystemofngraybodiesinnonemittingandnonabsorbing(nonparticipating)medium.Ingeneralcase,everynodereceivesdirectradiationfromremainingn1nodes,aswellasreflectedradiationfromithnodeofalln1nodes(includingit'sownreflectedradiation).L#LIfemissionintensitycanbeconsidereddirectionindependentandifreflectedradiationisnottakenintoaccount,geometricalaspectofradiationheatexchangesystemcanbecalculatedbymeansofviewfactorsM[1].ViewfactorMi,kdenotesthepartofradiationfromnodesurfaceiwhichfallsonnode (b"% surfacekrelativetowholeradiationenergyleavingnodei.ForeverynodesurfaceAi: )N#& 2&45VFBz (7% 0@Xdddddddd@E*dd!Pa*g| ##     ##Withtheaboveassumptions,radiationenergybalanceinunittimeforanynodecanbestatedas: `.'+ МIr20qrm]MI (7% p@Xdddddddd@rII    IKIr20qrm]MI (7 p@Xdddddddd@rIKK20qrm]MI (7l p@Xdddddddd@EvddvPyv| ##    ##K;;where i, kareemissioncoefficients,C0isradiationconstantandAi,Akaresurfacespertinentto &  nodesiandk.FornodeswhicharenotopticallyconnectedMi,k=Mk,i=0.WithAiMi,k=AkMk,i   Eq.(7)becomes:;Ӝ;J2#$hWGCz (7% p@Xdddddd@E %da fJJ2#$hWGCz (7 p@Xdddddd@E !d PB | ##    ##JTheaccuracyoftheaboveapproximationincreaseswithnodalsurfacetemperaturesbeingmoreuniformandwiththeiremissioncoefficientsbeingclosertounity.#XXX X!#;XXXX#XXX;X/# XXXX-.xZJFz|1 O ` @ HKK@EZ[ P ##                 ###XXX XH/#<XXXXSurfacetemperaturemeasurementerrorsateachnode(Fig.1)aremodeledwithemphasisonconvective b$  strayloosesandlocalthermocoupleinducedheatsinkaccordingtoHenneckeandSparrow[3].Theenergybalanceforunitlengthofthermocoupleleadisstatedthroughonedimensionalfintheoryas:#XXX<X2# XXXX B&" 2AHI\ LHzx7\% p@Xdddddddd@E'GddV-a2'f  ##     ###XXX X3#=XXXXwithAbeingthermocouplecrosssectionareaandRbeingthermocouplewirelateralthermalresistance +v%( perunitlength:#XXX=X#5# XXXX ,f&) МW2JKLV FBz (7% 0@Xdddddddd@E-dga-f WW2wyLW GCz (7% p@Xdddddddd@E-dda-f:  ##    0,*- k*TRW6&A4W6&A4 6&A4 6&A4T  WWWKk*TRW6&A4W6&A4 6&A4 6&A4T  k*TRW6&A4W6&A4 6&A4 6&A4T  k*TRW6&A4W6&A4 6&A4 6&A4T  k*TRW6&A4W6&A4 6&A4 6&A4T  KJk*TRW6&A4W6&A4 6&A4 6&A4T  k*TRW6&A4W6&A4 6&A4 6&A4T  JIk*TRW6&A4W6&A4 6&A4 6&A4T  IHk*TRW6&A4W6&A4 6&A4 6&A4T  H@k*TRW6&A4W6&A4 6&A4 6&A4T  @Wk*TRW6&A4W6&A4 6&A4 6&A4T  k*TRW6&A4W6&A4 6&A4 6&A4T  W#XXX X5#QXXXXBoundaryconditionscanbestatedas:#XXXQXI# XXXX v U2stOW GCz (7% p@Xdddddddd@Efddkaff:  ##    ##UU2opOW GCz (7% p@Xdddddddd@E. dga. f LLUL2f|OW GCz (7% p@Xdddddddd@E. dga. f ߜL#XXX XI#]XXXXApplicationofFourier'slawtoclassicsolutionofEq.(9)yieldsrateofheattransferthroughareaof .  contactbetweenthermocoupleandsurface:#XXX]XL# XXXX   V2uvRW GCz (7% p@Xdddddddd@E dd/ _a f  ##     ##VV2^_RW GCz (7% p@Xdddddddd@Edgaf ߜV#XXX XM#`XXXXThisheatratemustequalheatratesuppliedbywalltosamecircularareaofcontactwhichisstatedin R  [3]as:  B  #XXX`XO# XXXX 2UW GCz (7% p@Xdddddddd@Edd5 Paf|  ##   ###XXX XP#gXXXXwherefirsttermpertainstodisturbancetemperaturefieldandthesecondtermtotemperaturefieldin ~ solidintheabsenceoflocalsurfaceheatsink.Q*isdimensionlessheattransferrateassociatedwiththe n disturbancetemperaturefieldgivenin[3]asafunctionofBiotnumberofthesolid(Bi=hr/kw)for ^ 010ingraphicalform.#XXXgXQ# XXXXԜ N #XXX XS# XXXXԛ2" %_OKz| ^ pX@Xz@EqPznP2VqPIfTw(x)denotessurfacetemperatureintheabsenceofthethermocoupleandTj(x)temperatureatthe > junctionofthethermocoupleandsurface,E*(x)ismeasurementerrorduetopresenceofthermocouple: *  ##   ###XXX XT#aXXXXEquatingXEq.X(12)andYEq.YZYYZ(13)[,[\\]]dimensionlesstemperatureerrorE=E*/(TwT)canbestatedas: d #XXXaXgV# XXXX2VWXVFBz (7% 0@Xdddddddd@E8dd^ p&af ##    ##2YZ[VFBz (7% 0@Xddddddd@E#ddYa#f ##     ##m,  #XXX XvW#bXXXXwhere/representsmeasureofconductanceofthethermocouplecomparedtoconductanceofthe L(!$ channelwall.#XXXbX"Z# XXXXԀ#XXX XZ#;XXXXNonlinearequationsystemresultingfromEq.(1)andsubsequentequationshas <)"% followingmatrixform:YӜ^2M_OKz|7%U p@Xdddddd@E+U%5a+f^^2M_OKz|2 p@Xdddddd@E+_GPQ}+'T ##^̛̛̜ ##  where T isnodaltemperaturevector, K is#XXX;X)[# XXXXtemperaturedependent#XXX X]#;XXXXԀconductancematrixand P is -z'* #XXX;X^# XXXXtemperaturedependent#XXX X^#;XXXXԀnodalheatsource/sinkvector. .j(+ Nonlinearsystem,Eq.(17),requiresiterativesolutionconsistingoffollowingglobalsteps:R]ӄ0  Systemgeometryandnodalsource/sinkloadsareprescribed.0B*- # # c*TRW6&A4W6&A4 6&A4 6&A4T  ќ0  Initialguessofnodaltemperatures(i.e.vector T_)_ aabbbbbb`)`ccmustbegiven.v # # Є0  Heattransferresistancesbetweenthermallyconnectednodesarecalculated.f # # Є0  Equationsystemisformed,(nowtemperatureindependent,matrix K andvector P ).R # # Є0  Linearequationsystemissolvedyieldingvector T .B  # # Є0  Secondandsubsequentstepsarerepeated,withcalculatednodaltemperatures(when 2  necessary,withsomeunderrelaxation)usedasnextguess,untilsatisfactoryconvergence.  # #  ;QX;XL3  Resultsandconclusion#;XX;Qd#    L  Topresentthescopeofthemethoditisappliedtoestimationofsurfacemeasurementerrorsoninnerelectricalcoilsurfacesofcomplex,explosionprooftransformerusedtopowerminingventilationsystemsandotherpowerequipment.#XXX;X^# XXXXd{-6zE51r|09  ` @Ed   <XS {dd{-6PE51r|09  ` @Ed   4<XW {ߜdThetransformeriscontained(Fig.2)inexplosionproofenclosure -  whichisfilledwithair.It'sactive#XXX XKf#;XXXXԀ#XXX;Xh# XXXXpartconsistsof:threeleggedcoreandthreephasewindings#XXX X[h#;XXXXԀ#XXX;Xh# XXXXeach#XXX Xi#;XXXXԛ    ##              ##/e  Ӝwithtwolowvoltageandtwohighvoltagecoilsseparatedbyaxialcoolingducts.Beingthermallysymetric,onehalfofthetransformerismodelledby33nodethermalnetworkdepictedinFig.3.Jku-'i?/+p|0K `EL!K K" 8 @x!Fuu-'?/+p|0K `EL!K K" 8 t!Bu  Thedetailsregarding#XXX;Xbi# XXXXdesignofthermalnetwork,heattransferresistancesandempiricalexpressions#XXX Xxm#;XXXX  O dB*TRW6&A4W6&A4 6&A4 6&A4T  d ## "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# "#"# ##"#"#pertinenttodrytypetransformerscanbefoundin[4,5].Theresultingnonlinearequationsystemis q0)- B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  dB*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  dcB*TRW6&A4W6&A4 6&A4 6&A4T  cbB*TRW6&A4W6&A4 6&A4 6&A4T  baB*TRW6&A4W6&A4 6&A4 6&A4T  a`B*TRW6&A4W6&A4 6&A4 6&A4T  `_B*TRW6&A4W6&A4 6&A4 6&A4T  _^B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  B*TRW6&A4W6&A4 6&A4 6&A4T  ^]B*TRW6&A4W6&A4 6&A4 6&A4T  ]iterativelysolvedbymeansofdedicatedprogramsystem.Thetemperaturemeasurementerrorsforthe v caseofsurfaceattachedthermocouplesarecalculatedandpresentedinTable1forphaseBofthetransformerfornodesonthesurfacesoftheelectricalcoils.Pertinentcalculationparameterswere:coolingfluid:air,wallthermalconductivity:kw=0.2W/(mK)(aromaticpolyamideinsulatorlayer), :  thermocouplethermalconductivity:kTC=#XXX;Xn#17W/(mK),radiusofthermocouplewire:0.25mm,and &  surroundingairtemperature(node34):t=16C.2mӜӀ̛g*wT\ddd Xdd Xdd X##w,, , ,,V,,+  1" t  (1& &XXNode##XX&& # 9* &" 9Coil  &  PhaseB 9*  " 9& &XXC& &&& TW,C 9* & " 9Bi=hr/kw 9* & " 9E#CXX&C& ߘ#C~XCX  &  (Eq.15)#CXXC~љ#C& &XCX 9*r " 9E*,C(#CXX&C& /#C~XCXEq.14)#CXXC~#C& &XCX =*r " =TJ,C#CXX&C& #C~XCX pa* &" (   @2(@p2 XI>&"  @2 @ XLV1 9*&" 9126.7 eV&" $~?0.013175G$~?eМ1.32e-02 xK&"G $~?0.013175 $~? 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Q?0.28Q?0.28 nAG" Q?0.28 Q? %@10.5%@10.5 pAH" %@10.5 %@ Gzd[@109.57Gzd[@109.57#~C~##XX~#^TRI" Gzd[@109.57 (  Gzd[@ ^М& &XXLӀ ^I L #XX&& #~XXLLTable1 Calculatedtemperaturemeasurementerrorsforthermocouplesattached J МtothesurfaceofphaseBwindingsof315KVAexplosionprooftransformer.#XX~s#~XXԛ dK #XX~#L  LӜ XXXXLӀ (L L9Measurementofsurfacetemperaturesisanecessarystepinexperimentaldesignstagesofpowertransformers,wheretemperatureextremainfluenceinsulationlifetime.Surfacemountedthermocouplesareusedasarulesinceelectricalconsiderationsorotherreasonsdonotpermitwallpenetration.Errorsemanatingfrominterfacecontact,junctionbeingdisplacedfromthesurface,attachmentsubstratetypesetc.,arenotemphasizedinthispaper.Theydependonthermocouplemountingtechniqueandcanbemodeledregardlessofthecomplexityoftheapparatus.#XXX X# XXXXԀErrorsemanatingfromstrayconvectiveheatloss  R fromhotjunctionandfromtheleads,causeperturbationofthesurfacetemperatureanddependonheattransferconditions.#XXX Xպ# XXXXԀIncaseofcomplexsystemsapplicationofnetworkanalogycanprovideforerror XT estimationasseenfromexampleinTable1.Thosearemeanerrorsforagivensurfaceandthereforeonlyorientational.Estimationoflocalsurfacetemperaturemeasurementerrorshavetobeperformedsubsequentlybycontinuumapproximationofsomesmallpartoftheapparatus,likeforexample.in[6]#XXX Xػ# XXXX, !W withboundary#XXX X# XXXXconditionstakenfromtheglobalnetworkanalogymodel. "X #XXX X# QXX4  References#XXQ:#  #jZ Мk~XX :-~Ӏ $x[ $fx#~- :#e5~[1]0  EckertE.R.G.,Drake,R.:,AnalysisofHeatandMassTransfer,McGraw,NY,1972.R%\ # # МfLӀLUxz[2]0  Isachenko,V.P.,Osipova,V.A.,Sukomel,A.S.:Heattransfer,MIR,Moscow,1977.V&^ # # МLӀL2Z|[3]0  Hennecke,D.K.,Sparrow,E.M.:LocalHeatSinkonConvectivelyCooledSurfaceApplicationtoSurface Z' ` MeasurementError,Int.J.HeatMassTransfer,vol13,pp.287304,1970.  # # М^LӀLW~[4]0  Lein,P.:BeitragzurErwrmungsberechnungvonTrockentransformatorenmitaxialenKhlkanlen,Elektrie,5, &)"c 1967.  # # МLӀL[5]0  Zvizdic,D.:Numericalcalculationofminingtransformertemperaturerise,Elektrotehnika,ELTHB2 *|$f 27(1984)5,pp.311-320,Zagreb.(inCroatian).  # # МLӀLH[6]0  Zvizdic,D.:CalculationofSurfaceTemperatureMeasurementErrorsinbuoyantChannelFlow,Proceedingsof ,H&i М5thInternationalSymposiumonTemperatureandThermalMeasurementinIndustryandScienceTEMPMEKO -'j 93,pp.112116,ref.4,Prague,1993.  # # #eXX5e#e5XeX 3Ӏ /(l 3Addressoftheauthors  /+)m 0  #eXX5eϝ#e5XeXProf.dr.DavorZvizdic,dipl.ing.,Prof.dr.IvanGalaso,dipl.ing.,FacultyofMechanicalEngineeringandNaval i0)n Architecture,UniversityofZagreb,IvanaLucica5,10000Zagreb,Croatia,email:davor.zvizdic@fsb.hr#~5eϟ##XX~q#ԛ